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Properties

Label 627.2.i.e
Level $627$
Weight $2$
Character orbit 627.i
Analytic conductor $5.007$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [627,2,Mod(463,627)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(627, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("627.463");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.i (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 12 x^{12} - 5 x^{11} + 94 x^{10} - 34 x^{9} + 375 x^{8} - 46 x^{7} + 1040 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{10} - 1) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{3}) q^{4} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{2} + 1) q^{7}+ \cdots - \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 7 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} + 14 q^{7} + 12 q^{8} - 7 q^{9} - 13 q^{10} - 14 q^{11} + 18 q^{12} - 8 q^{13} - 16 q^{14} - 2 q^{15} - 5 q^{16} - 9 q^{17} + 2 q^{18} - 14 q^{19} - 32 q^{20}+ \cdots + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 12 x^{12} - 5 x^{11} + 94 x^{10} - 34 x^{9} + 375 x^{8} - 46 x^{7} + 1040 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12492522413 \nu^{13} - 204368968522 \nu^{12} + 542757010127 \nu^{11} + \cdots + 23712717353469 ) / 229458856979425 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 191876446109 \nu^{13} - 392846741171 \nu^{12} + 2290573490111 \nu^{11} + \cdots - 688264138236558 ) / 229458856979425 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 233645909396 \nu^{13} + 1159637467626 \nu^{12} - 858485307716 \nu^{11} + \cdots - 78464232577577 ) / 229458856979425 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1254063459263 \nu^{13} + 2622520781297 \nu^{12} - 21134211907002 \nu^{11} + \cdots - 5542565905719 ) / 688376570938275 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1437608426327 \nu^{13} + 2093175839863 \nu^{12} - 16568600828333 \nu^{11} + \cdots + 400858213276249 ) / 458917713958850 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1050777220112 \nu^{13} + 3824944080053 \nu^{12} - 16370513374848 \nu^{11} + \cdots + 931396499850044 ) / 229458856979425 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3414237642421 \nu^{13} - 7699374314749 \nu^{12} + 43882057476759 \nu^{11} + \cdots - 25\!\cdots\!27 ) / 458917713958850 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14394158081609 \nu^{13} + 19182465543746 \nu^{12} - 173373160679511 \nu^{11} + \cdots - 12\!\cdots\!92 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7904239117823 \nu^{13} - 7941716685062 \nu^{12} + 95463976319442 \nu^{11} + \cdots + 42588388412874 ) / 688376570938275 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7904239117823 \nu^{13} - 7941716685062 \nu^{12} + 95463976319442 \nu^{11} + \cdots + 730964959351149 ) / 229458856979425 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 38616818188369 \nu^{13} - 35376655705411 \nu^{12} + 457197545552226 \nu^{11} + \cdots + 29\!\cdots\!22 ) / 688376570938275 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 93208391901109 \nu^{13} + 95995182831046 \nu^{12} + \cdots - 79\!\cdots\!92 ) / 13\!\cdots\!50 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 3\beta_{10} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} - \beta_{12} - 6 \beta_{11} + 13 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{13} - 9\beta_{12} - 2\beta_{11} + 10\beta_{9} - \beta_{5} + \beta_{4} - 28\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -21\beta_{8} - 12\beta_{7} - 23\beta_{6} + 36\beta_{3} - 16\beta_{2} + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 80 \beta_{13} + 68 \beta_{12} + 26 \beta_{11} - 11 \beta_{10} - 82 \beta_{9} - 80 \beta_{8} + \cdots + 170 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 177 \beta_{13} + 109 \beta_{12} + 225 \beta_{11} - 347 \beta_{10} - 201 \beta_{9} + 3 \beta_{5} + \cdots - 347 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 600\beta_{8} + 491\beta_{7} + 630\beta_{6} - 89\beta_{4} - 252\beta_{3} + 1093\beta_{2} - 173 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1393 \beta_{13} - 902 \beta_{12} - 1465 \beta_{11} + 2039 \beta_{10} + 1601 \beta_{9} + \cdots - 1535 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4409 \beta_{13} - 3507 \beta_{12} - 2179 \beta_{11} + 1844 \beta_{10} + 4717 \beta_{9} - 649 \beta_{5} + \cdots + 1844 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -10656\beta_{8} - 7149\beta_{7} - 12262\beta_{6} + 561\beta_{4} + 9869\beta_{3} - 12778\beta_{2} + 12776 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 32226 \beta_{13} + 25077 \beta_{12} + 17760 \beta_{11} - 16756 \beta_{10} - 34954 \beta_{9} + \cdots + 50450 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/627\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(419\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{10}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
463.1
1.35831 − 2.35267i
0.914361 − 1.58372i
0.794699 − 1.37646i
−0.0560088 + 0.0970101i
−0.371378 + 0.643245i
−1.01277 + 1.75417i
−1.12721 + 1.95239i
1.35831 + 2.35267i
0.914361 + 1.58372i
0.794699 + 1.37646i
−0.0560088 − 0.0970101i
−0.371378 − 0.643245i
−1.01277 − 1.75417i
−1.12721 − 1.95239i
−1.35831 − 2.35267i −0.500000 − 0.866025i −2.69004 + 4.65928i −0.467652 − 0.809997i −1.35831 + 2.35267i 4.17977 9.18241 −0.500000 + 0.866025i −1.27044 + 2.20046i
463.2 −0.914361 − 1.58372i −0.500000 − 0.866025i −0.672111 + 1.16413i 1.20135 + 2.08079i −0.914361 + 1.58372i 2.48304 −1.19924 −0.500000 + 0.866025i 2.19693 − 3.80519i
463.3 −0.794699 − 1.37646i −0.500000 − 0.866025i −0.263092 + 0.455690i −1.98856 − 3.44428i −0.794699 + 1.37646i 0.0905430 −2.34248 −0.500000 + 0.866025i −3.16061 + 5.47434i
463.4 0.0560088 + 0.0970101i −0.500000 − 0.866025i 0.993726 − 1.72118i −1.02085 − 1.76816i 0.0560088 − 0.0970101i −3.03154 0.446665 −0.500000 + 0.866025i 0.114353 − 0.198065i
463.5 0.371378 + 0.643245i −0.500000 − 0.866025i 0.724157 − 1.25428i −1.26435 − 2.18992i 0.371378 − 0.643245i 4.12826 2.56125 −0.500000 + 0.866025i 0.939105 − 1.62658i
463.6 1.01277 + 1.75417i −0.500000 − 0.866025i −1.05142 + 1.82111i 1.77847 + 3.08040i 1.01277 − 1.75417i 3.36717 −0.208293 −0.500000 + 0.866025i −3.60237 + 6.23949i
463.7 1.12721 + 1.95239i −0.500000 − 0.866025i −1.54123 + 2.66948i 0.761596 + 1.31912i 1.12721 − 1.95239i −4.21725 −2.44032 −0.500000 + 0.866025i −1.71696 + 2.97387i
562.1 −1.35831 + 2.35267i −0.500000 + 0.866025i −2.69004 − 4.65928i −0.467652 + 0.809997i −1.35831 − 2.35267i 4.17977 9.18241 −0.500000 − 0.866025i −1.27044 − 2.20046i
562.2 −0.914361 + 1.58372i −0.500000 + 0.866025i −0.672111 − 1.16413i 1.20135 − 2.08079i −0.914361 − 1.58372i 2.48304 −1.19924 −0.500000 − 0.866025i 2.19693 + 3.80519i
562.3 −0.794699 + 1.37646i −0.500000 + 0.866025i −0.263092 − 0.455690i −1.98856 + 3.44428i −0.794699 − 1.37646i 0.0905430 −2.34248 −0.500000 − 0.866025i −3.16061 − 5.47434i
562.4 0.0560088 − 0.0970101i −0.500000 + 0.866025i 0.993726 + 1.72118i −1.02085 + 1.76816i 0.0560088 + 0.0970101i −3.03154 0.446665 −0.500000 − 0.866025i 0.114353 + 0.198065i
562.5 0.371378 − 0.643245i −0.500000 + 0.866025i 0.724157 + 1.25428i −1.26435 + 2.18992i 0.371378 + 0.643245i 4.12826 2.56125 −0.500000 − 0.866025i 0.939105 + 1.62658i
562.6 1.01277 − 1.75417i −0.500000 + 0.866025i −1.05142 − 1.82111i 1.77847 − 3.08040i 1.01277 + 1.75417i 3.36717 −0.208293 −0.500000 − 0.866025i −3.60237 − 6.23949i
562.7 1.12721 − 1.95239i −0.500000 + 0.866025i −1.54123 − 2.66948i 0.761596 − 1.31912i 1.12721 + 1.95239i −4.21725 −2.44032 −0.500000 − 0.866025i −1.71696 − 2.97387i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 463.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 627.2.i.e ✓ 14
19.c even 3 1 inner 627.2.i.e ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.i.e ✓ 14 1.a even 1 1 trivial
627.2.i.e ✓ 14 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(627, [\chi])\):

\( T_{2}^{14} + T_{2}^{13} + 12 T_{2}^{12} + 5 T_{2}^{11} + 94 T_{2}^{10} + 34 T_{2}^{9} + 375 T_{2}^{8} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{14} + 2 T_{5}^{13} + 26 T_{5}^{12} + 32 T_{5}^{11} + 429 T_{5}^{10} + 507 T_{5}^{9} + \cdots + 62500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 62500 \) Copy content Toggle raw display
$7$ \( (T^{7} - 7 T^{6} + \cdots - 167)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 8 T^{13} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{14} + 9 T^{13} + \cdots + 242064 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 893871739 \) Copy content Toggle raw display
$23$ \( T^{14} + 4 T^{13} + \cdots + 24964 \) Copy content Toggle raw display
$29$ \( T^{14} + 18 T^{13} + \cdots + 4080400 \) Copy content Toggle raw display
$31$ \( (T^{7} + 11 T^{6} + \cdots + 2272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 17 T^{6} + \cdots + 25138)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 11 T^{13} + \cdots + 25887744 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 36773431696 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 521756964 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 566059264 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 1139737600 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 20194683664 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 143747139600 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 22895650803600 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 67602080016 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 3037686466816 \) Copy content Toggle raw display
$83$ \( (T^{7} - 9 T^{6} + \cdots + 316262)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 1942042919184 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 5784058809 \) Copy content Toggle raw display
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